Calculus (4)

• Mathematics
• Calculus

In this final module of Calculus (MATH4009), we will develop calculus on “vector fields”. Vector fields are vector-valued functions arise naturally from Physics and we will discuss how to make sense of integrals of them over curves and surfaces. Topics to be discussed include: - Line integrals and Green’s Theorem - Conservative of vector fields - Surface integrals and Flux - Stokes’ and Divergence Theorem In particular, Green’s, Stokes' and Divergence Theorem can be regarded as a vast generalisation of the Fundamental Theorem of Calculus, for line and surface integrals. As an application, we will derive the Gauss' Law that describes the flux of an inverse square field across a closed surface. 

 Finally, to complete the discussion on limits of a function or a (infinite) sum of functions in the course of the study of Calculus, the definitions of limits of sequences and series are also introduced, which provide the theoretical basis of the introduction of a “power series”. “Power series” is a generalization of polynomials and can be used to represent elementary as well as more general functions, which paves the way for more advanced analysis of functions, necessary in practical applications. Definitions are discussed and the most important theorems are derived in the lectures with a view to help students to develop their abilities in logical deduction and analysis.